B R I N G I N G O U T T H E A L G E B R A I C

C H A R A C T E R O F A R I T H M E T I C

D A V I D C A R R A H E R , T E R C A N A L Ú C I A D . S C H L I E M A N N , T U F T S U N I V E R S I T Y B Á R B A R A M . B R I Z U E L A , H A R V A R D U N I V E R S I T Y

*First Draft of a paper presented at symposium, “Toward a research base for algebra reform beginningin the early grades” during the Annual Meeting of the American Educational Research Association, Montreal,Canada, April 21, 1999. Partial support was provided by NSF Grant #9722732, "Bridging Arithmetic andAlgebra", awarded to Analúcia D. Schliemann. Support for the multimedia version of this paper was providedby NSF-Grant 9805289, "Bridging Research and Practice", awarded to Ricardo Nemirovsky & DavidCarraher. Special thanks go to Thelma Davis and John F. O'Meara of East Somerville Community School,Somervile, MA and to Cara DiMattia of TERC.

Comments to david_carraher@pop.terc.edu; aschliem@tufts.edu; brizueba@gse.harvard.edu

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B R I N G I N G O U T T H E A L G E B R A I C C H A R A C T E R O F A R I T H M E T I C

E A R L Y A L G E B R A

For many years, educational research about algebra highlighted the considerabledifficulties adolescents display in understanding and using algebra (Booth, 1984; DaRocha Falcão, 1992; Filloy & Rojano, 1989; Kieran, 1985a, 1989; Laborde, 1982;Resnick, Cauzinille-Marmeche, & Mathieu, 1987; Sfard & Linchevsky, 1984; Steinberg,Sleeman & Ktorza, 1990; Vergnaud, 1985; Vergnaud, Cortes, & Favre-Artigue, 1987;Wagner, 1981). These difficulties were not surprising for those who tended to think ofalgebra as requiring the emergence of formal operational thinking. Regardless of one’stheoretical stance, the research findings alone seemed to justify delaying algebrainstruction until students were about 12 years old and they had already established asolid grounding in arithmetic. To help children overcome the difficulties encountered inthis transition from arithmetic to algebra, researchers (e.g., Herscovics & Kieran 1980;Kieran, 1985b) developed teaching approaches to help seventh and eighth gradestudents use their knowledge of arithmetic to understand algebraic equations.

If adolescents were apparently having so many difficulties with algebra, what couldpossibly recommend early algebra? A quick look at some empirical findings andrecommendations can shed light on this matter.

Developmental studies reveal that even seven year-olds understand the basic logicof equations when grounded in reasoning about quantities (Carpenter & Levi, 1998;Schliemann, Brito-Lima, & Santiago, 1992; Schliemann, Carraher, Pendexter & Brizuela,1998). Additionally, children in elementary school classrooms were found to usealgebraic reasoning while they interact with their peers and the teacher to solve relativelycomplex, open-ended problems (Schifter, 1998). Furthermore, there has been at leastone eminently successful attempt to integrate algebra into mathematics teaching andlearning beginning from the first grade (Bodanskii, 1991). In Bodanskii’s study, fourthgraders in the Soviet experimental group used algebra notation to solve verbalproblems. These fourth grade students not only performed better than their controlpeers throughout the school years, but also showed better results in algebra problem-solving when compared to sixth and seventh graders who followed the traditional path offive years of arithmetic instruction followed by algebra instruction from grade six on.

But should we wait until children are 12 years old to focus on the logical relations?One of the main features that characterizes additive structures suggests not. Piagetcharacterized a structure as a system of relations, stressing that what are important arethe relations between elements, and not the elements themselves (Piaget, 1970/1968).Thus, the additive structures correspond to a set of relations between elements, and notonly elements, yields, or products. By focusing arithmetic instruction on the latter, weare ignoring one of the inherent characteristics of the number system and of the additive

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structures domain. Knowledge construction consists “in establishing relations,identifying interactions and constructing interconnections, with which the data providedby experience are organized” (Piaget, in García, 1997, p. 63). Arithmetic instructiontypically ignores the relations between elements and their transformations and fails torecognize the basic ways through which we organize experiences and constructknowledge. These arguments further question educational approaches that continue todichotomize arithmetic and algebra.

In the last fifteen years, some researchers have recommended an algebraicapproach during the elementary years of mathematics education. Davis (1985, 1989)argues that preparation for algebra should begin in grade two or three. Vergnaud (1988)suggests that instruction in algebra or pre-algebra should start at the elementary schoollevel so that students will be better equipped to deal later with the epistemological issuesinvolved in the transition from arithmetic to algebra. Kaput (1995) sees the traditionalalgebra curriculum as alienating students from genuine mathematical experience and asan engine of inequity. He further proposes that the integration of algebraic reasoningacross all grades is the key to add coherence, depth, and power to school mathematics,eliminating the late, abrupt, isolated, and superficial high school algebra courses. “Analgebrafied K-12 curriculum helps democratize access to powerful ideas.” (Kaput, 1995).

The task before educators goes far beyond embracing the general idea of earlyalgebra. For early algebra is not about teaching algebra, as we now teach it, to youngerstudents. It is about new ways of looking at how arithmetic and algebra are interwovenor, as we will attempt to illustrate here, at drawing out the algebraic character ofarithmetic. To do this represents a considerable departure from the present state ofaffairs. How should this be done? What kinds of problems should be used? What sortsof new demands does this place on teachers and students? What sorts of adjustmentsand new ideas in teaching and learning promise to be fruitful?

Addition and subtraction may provide fruitful contexts for exploring the algebraicnature of arithmetic. This is supported by an impressive series of Soviet studies inmathematics education (e.g., Bodanskii, 1991) in which children discussing,understanding, and dealing with algebraic concepts and relations much earlier than isthe norm nowadays. These intriguing developments have helped mathematics educatorsgradually realize that arithmetic and elementary algebra are far more intertwined thancommonly supposed1.

Here is where the research base for early algebra comes into play. As we notedabove, there is already a rich background of research on learning and development thatcan be drawn upon. One of the underlying themes in research has been that children’sunderstanding of how quantities combine to make new quantities and break apart intocomponent quantities may be crucial to much of mathematical learning. We stronglysuspect that variable quantities can play an important role in the emergence of

1 Bodanskii (1991) discusses the tension between arithmetical and algebraic methods for solving verbalproblems and concludes, “the algebraic method is the more effective and more ‘natural’ way of solvingproblems with the aid of equations in mathematics” (p. 276). We tend to view the issues as demanding anew conceptualization of arithmetic rather than its wholesale displacement by algebra. To be fair, this alsoseems more in keeping with what Bodanskii and colleagues have tried to achieve.

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mathematical variables and that using notation for describing the relations amongquantities foreshadows and paves the way for using notation about variables and theirinterrelations in functional notation. From this viewpoint, addition and subtraction can beapproached, from the start as additive functions amenable to description throughalgebraic notation.

In the present paper we will explore some of the issues young learners face in tryingto understand addition and subtraction from an algebraic standpoint. Let us begin toclarify what we mean here by "algebraic standpoint". In many classroom contexts, theexpressions, A + B = C and A = C - B, are taken as having different meanings: the first isabout adding, the second about subtracting2. This focus on arithmetical operations asactions one "does" (that presumably mirror events such as "lost", "won", "used up" instory problems) contrasts sharply with the view that the expressions describe relationsamong the quantities, A, B, and C. From this latter, algebraic standpoint, theexpressions "A + B = C" and "A = C - B" are interchangable; they express the very samerelations (albeit in slightly different ways)3. But much more is involved than how oneparses notation. That said, will be an attempt from here on to defer as much as possibleto what children themselves have to say (and write and gesture), for we have found thatthey often express, in telling ways, the issues they are coming to grips with. Indeed,wherever problems are consistently taken by children in ways differently than they aregiven by adults, there may be something important for us to learn.

T H E R E S E A R C H S E T T I N G

The present research was undertaken as part of an Early Algebra study of theauthors of this paper with a classroom of 18 third grade students at a public elementaryschool in the Boston area. The school serves a diverse multiethnic and racialcommunity reflected well in the class composition, which included children from SouthAmerica, Asia, Europe, and North America. We had undertaken the work to understandand document issues of learning and teaching in an "algebrafied" (Kaput, 1995)arithmetical setting. During the first semester of third grade, when the study heredescribed took place, the teacher had been addressing primarily issues of addition andsubtraction. Our activities in the classroom consisted of teaching a two-hour “mathclass” on a weekly basis, for a total of five weeks. In addition, during the following threeweeks we interviewed children about issues related to the content of the lessons. Theinterviews were treated as an additional source of data as well as an opportunity for thechildren to develop a greater understanding of problems similar to those presented inclass. The topics for the class sessions evolved from a combination of the curriculumcontent, the teacher's main goals for the semester and the questions the researchers

2 Indeed, the latter expression may be viewed as altogether incorrect by some children, for whom the“problem” should be on the left side and the “answer” on the “gives” side.

3 These are not totally exclusive viewpoints. Even an additive comparison problem (Tom has $4.00more than Maria) is amenable to expression as an implicit or virtual action (e.g., If we were to take away$4.00 from Tom or give $4.00 to Maria, they would have the same amounts).

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brought to the table. For example, as the study began, the teacher was reviewing withthe children the topics of "more" and "less" and the use of notation for "greater than" and"less than". We were well aware of the relative difficulty of additive comparisonproblems (Carpenter & Moser, 1982; Vergnaud, 1982) and sensed from the start thatadditive comparisons would provide a special opportunity for approaching arithmeticfrom an algebraic standpoint. This approach is similar in several regards to Thompson's(1991) teaching experiments with fifth grade students.

From a research only viewpoint, we would normally wish to follow one or two childrenin order to trace their thinking over time. But since we were teaching simultaneously aswe were doing research, the researchers, who also worked holding cameras, oftenfound themselves divided as helpers in the class activities and as documentarists. Forthese reasons, it is necessary to tell part of our story with some children, and part withothers. We are reasonably confident that the great majority of students were confrontingsimilar issues. Among those issues was that of recognizing, for example, that theproblem we focused on entailed differences between heights, which many of the childreninitially interpreted as heights of individual children in the problem.

T H E P R O B L E M

Here is the problem we devised for the fourth week's class. We had already spent threeweeks working with vector-like representations for quantities similar to the directed linesegment next to Maria. In the first two weeks they showed us that they spontaneouslyrepresented amounts through iconic representations (E.g., candies drawn in paperwrappers, fish as icons of fish, with eyes, fins, scales, etc.), and seemed to prefer themto our suggestions to use vectors. Gradually, however, they began to use "arrows" torepresent both discrete and continuous quantities and would with a certain delight shiftdiscussion from one context to the other (e.g., fish dying in a fishbowl, to spendingmoney or consuming cookies) for the same configuration of arrows. This, incidentally,attests somewhat to the usefulness of generic referents for quantities over iconicrepresentations, which are not easily interchangeable.

At the beginning of the fourth class, the following problem was shown to the children viaa projector onto a large screen. Some of the students read the sentences aloud as thefirst author read it to them.

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F i g u r e 1 : A p r o b l e m o f a d d i t i v e d i f f e r e n c e s

Q U A N T I T I E S A N D T H E I R D I F F E R E N C E S

The children clearly understood that the problem referred to the heights of threechildren, and that there was some variation in these heights. However, it was apparentthat many of the students were thinking about the numbers in ways quite differently fromus. Claudia decided that Maria was taller than Tom and then changed her mind. Sheand her classmates were treating the numbers as total heights. (“Tom is four inchestall…”). Nathia decided that Maria must be six inches tall and Leslie was 12 inches tall.Michael volunteered that Tom was 10 inches tall (see Transcript 1 and Clip 1 below),apparently having added the two numbers given in the problem. In order to make surethe children were clear about the magnitude of the measures, we pulled out a measuringtape and showed them how wide an interval 4 inches would span. Most childrenimmediately tried to separate their two palms by that amount, as if to calibrate theamount in a personally meaningful way. However, although children recognized that 10inches was inordinately short for a child, they continued to view the measures in theproblem as referring to absolute heights as shown in the discussion with Michael(Transcript 1).

T r a n s c r i p t 1 : M i c h a e l a f f i r m s h i s i n i t i a l r e a d i n g o f t h e p r o b l e m . Ma n y o t h e r s t u d e n t s a p p e a r t o a g r e e w i t h h i m .

1 0 I N C H E S T A L L

David: … [Tom] must be about this big right?

� Tom is 4 inches taller than Maria.

� Maria is 6 inches shorter than Leslie.

� Draw Tom’s height, Maria’s height, and Leslie’s height.

� Show what the numbers 4 and 6 refer to.

Maria Maria’sheight

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Michael: [Yeah..] no, Tom, Tom is…: Ten inch…

David: … must be a puppet. Then why did you, why did you blurt out, whydid you say very quickly that Tom was ten inches? I have a feeling Iknow why you said it.

Michael: Maria… She is six inches…

David: It says Maria is six inches tall?

Michael: But I don't mean she's this small.

David: No? Well, what does it say there, Michael? Read it. Read thesecond sentence for us.

Michael: Maria is six inches shorter than Leslie.

David: Than Leslie, right. So, is anybody six inches tall in this problem?

Michael: Yes.

David: Who?

Michael: Maria. (others agree)

David: Maria? She's six inches tall? Is that what the problem says?

Michael: Yeah…

C l i p 1 : M i c h a e l t r y i n g t o s h o w a h e i g h t o f 6 i n c h e s ( w h a t h e t a k e s a s M a r i a ' s h e i g h t ) .

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We then decided to enact the problem in front of the class with three volunteers.Jessica was chosen to represent Tom. With some discussion, the children agreed thatthe stand-in for Maria (Jennifer) should be shorter than Jessica (Tom). Finally, Reynoldwas chosen to represent Leslie; he was noticeably taller than the other two. In thecourse of this acting out, the three actors found it helpful to use their hands as we rereadthe problem and verified whether we had the information straight. We then asked thestudents to show the differences in height—e.g. “how much taller is Tom than Maria.”Faced with this question, the children sometimes pointed to one child, sometimes to theother. At our insistence to show “how much more,” they preferred to show one heightand then the other by placing a hand on each respective head, either in sequence orsimultaneously. In retrospect, it appears that the children were not inclined to view therelative differences in heights as entities that occupied a region in space. It is as if“more” was not a bona fide amount, but rather a state of affairs that required reference totwo legitimate amounts, namely the two children referred to in the comparison. However,Jennifer clearly identified an interval above her head to represent how much taller “Tom”was than her.

In this acting out, the members of the class became convinced that they now had theorder of the heights straight. And indeed they did. Leslie was the tallest; Maria was theshortest, Tom was in between. When asked the trick question of whether Tom was“taller or shorter?” some students voiced support of one position, others supported the

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other. Then they quickly realized that Tom was taller (than Maria) and shorter (thanLeslie) at the same time.

D R A W I N G T H E C H I L D R E N ' S H E I G H T S

Now that the children were fairly comfortable with the premises of the problem, weasked them to draw the heights of the three protagonists on a piece of paper using“arrows.” As mentioned before, they had already practiced using vector-like symbols(“arrows”) in earlier sessions. We were curious about how they would include thevalues, 4 inches and 6 inches, in their drawings. Since we had collectively settled theissues of the relative heights, not surprisingly 15 of the 16 children in the class that daywho made drawings got the orders of the heights correct. Even though they openlycompared and discussed their drawings, there was a fair degree of variability in theirapproaches. In a continuum from iconic to use of arrows in a comparative fashion, twochildren produced iconic drawings of Maria, Tom, and Leslie (with hands, feet hair, etc.)and 4 more, like Samantha, drew both arrow diagrams and iconic pictures (See Figure2).

F i g u r e 2 : S a m a n t h a ' s i n t e r p r e t a t i o n o f t h e s t u d e n t ' s h e i g h t s . T h e i n s t r u c t i o n s , t h e s a m e a s i n F i g . 3 , a r e n o t s h o w n .

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Two other children drew a person-arrow hybrid, that is, vertical lines with a head ontop; 8 children simply drew arrows to represent the children or their heights as inMelissa’s case (Figure 3).

F i g u r e 3 : M e l i s s a ' s r e n d e r i n g o f t h e s t u d e n t ' s h e i g h t s .

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Yasmeen drew one vertical line with the letters T, L, and M labeling distinct heightsalong the line (see Figure 4). Incidentally, this child hardly ever participates in classroomdiscussion and tends to work in isolation, not always following the prescribed activities.

F i g u r e 4 : Y a s m e e n ' s r e n d e r i n g o f t h e s t u d e n t ' s h e i g h t s .

Twelve of the 16 pupils used numbers in their drawings. Although only 2 numberswere given in the problem, every student who used numbers provided three, one eachfor Tom, Maria, and Leslie. (Actually, in two cases children drew a series of numbersnext to the children’s height-arrows as shown in Samantha’s drawing (Figure 2). In 7 ofthe 12 cases where numbers were used, children provided answers that were consistentwith the information given in the problem. Interestingly, however, none of our studentsspontaneously used the values 4 inches and 6 inches explicitly in their drawings. It isnot that they did not make use of this information; as we said, 7 children gave totalheights that embodied these differences. But they did not label any parts of heights orintervals between heights as corresponding to these values.

It is also interesting to note that the greatest height given by a student was 16inches. Since in the course of our earlier discussion we had already noted that thechildren in the class were in the neighborhood of 48 inches tall, I asked them to explainwhy the numbers were all so small. Melissa wittingly volunteered that Tom, Maria, andLeslie were probably 3 dolls!

Kevin’s dialogue with Barbara shows how he worked out the differences in heightbetween Leslie and Tom and, when prompted, how he attempted to hypothesize aheight for Maria.

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T r a n s c r i p t 2 : K e v i n d e d u c e s t h e d i f f e r e n c e b e t w e e n L e s l i e a n d T o m ' s h e i g h t s . H e h a s s p e c u l a t e d a b o u t M a r i a ' s h e i g h t .

R E L A T I V E H E I G H T S

Bárbara: So show us, …

Kevin: So she's (Leslie) taller than Tom by two inches. Mmhumm… cos Tom's tallerthan Maria by four inches… Mmhumm… and Maria is taller than, and Leslieis taller than Maria by six inches so that means that if he's taller than her byfour inches and she, then Leslie has to be taller than Tom by two inches.

Bárbara: Mm… and to figure that out do you need to know exactly how high each oneis?

Kevin: Yes

Bárbara: You do? How did you do it if you don't know how high they are?

Kevin: I do... cuz Maria's... Maria's…

Bárbara: Tell me exactly how high they are.

Kevin: Maria's... four feet six inches… and uh…

Bárbara: Where did you get that from?

Kevin: I don't know it just looks like that.

Bárbara: I think, I think you're just inventing that.

Kevin: I am.

Bárbara: You are.

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C l i p 2 : K e v i n e x p l a i n i n g h i s l o g i c .

I N F E R R I N G A N D V I S U A L L Y I N D I C A T I N G T H E D I F F E R E N C E S

After the students had prepared their own drawings, we showed the diagram inFigure 5 and asked them to explain their answers. Typically, a student would point tothe respective rectangle and read off the height. Children seemed to treat their answersas “the correct” answers rather than one correct set among many possibilities. Whenasked how they knew Leslie was 12 inches tall, they would justify their answers byreferring to the relative information (“because Maria is 6 inches shorter”). Apparently,none of the children realized that neither child’s height was determined by theinformation given in the problem.

It was revealing how children attempted to relate the numbers 4 and 6 to thediagram, when prompted. We expected that since they had often used the numberscorrectly in their drawings, they would easily isolate the parts of heights by which oneexceeded the other or fell short of the other. However, this did not occur. As they haddone before when comparing actual children in the room, they tended to point to thehighest part of one or both “children”, i.e., the rectangles in the diagram. When wecontinued asking them to show us something that corresponded to 4 inches, theychanged from marking off a vertical line within a rectangle to sweeping out a region.

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Some children would highlight the region between two rectangles. Others would refer toa vertical region. (A more careful analysis of the videotape will be provided in the finalversion of this paper).

F i g u r e 5 : D i a g r a m f o r a s h o r t w h o l e c l a s s d i s c u s s i o n .

With encouragement, several students came to the realization that we could labelvertical regions or parts of rectangles as corresponding to 4 and 6 inches. Thisdiscovery seemed to be crucial in the children’s coming to understand differences asquantities that can be visually expressed in a consistent manner. Here is how Sarahshows 6 as the difference between two heights:

T r a n s c r i p t 3 : S a r a h p o i n t s t o a n a d d i t i v e d i f f e r e n c e . .

W H E R E I S T H E 6 ?

David: She's (Maria) six inches shorter... Where does that six go? If you hadto point to something up here (indicating projection of Figure 5 on thescreen), Sarah, could you show me what six refers to? …Go ahead.(To the other students) Take a look at what Sarah's going to do now.

Sarah: (Pointing to the difference between Maria and Leslie) This space righthere

David: See this space right here. That's kind of the difference…

Actually, Sarah's gesture has three parts. First she sweeps her hand from the top ofMaria's height (line segment) horizontally rightward to the corresponding height againstLeslie's line segment. Then she sweeps her hand upwards indicating the region bywhich Leslie's height surpasses that of Maria. Finally, she sweeps out a diagonal fromthe top of Leslie's height to the top of Maria's height. The three movements comprise a

Tom Maria LeslieTom Maria Leslie

• Tom is 4 inches taller than Maria.

• Maria is 6 inches shorter than Leslie.

• Show what the numbers 4 and 6 refer to.

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triangle. In accepting her answer, David highlights the second of the two movements.Incidentally, the same question sometimes evinced slightly different responses fromother students. Erica, for example, restricts herself to the first of Sarah's threemovements to indicate where the "6" is. Other students make only the diagonalmovement. It is not altogether clear what these individual variations mean but togetherthey suggest that some students feel that it is necessary to include both elements (eachchild's height) in expressing the difference.

C l i p 3 : S a r a h i n d i c a t e s t h e r e g i o n o f L e s l i e ' s h e i g h t t h a t c o r r e s p o n d s t o t h e a m o u n t b y w h i c h s h e s u r p a s s e s M a r i a ( 6 i n c h e s ) .

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E X P R E S S I N G R E L A T I V E D I F F E R E N C E S T H R O U G H M O R E G E N E R A L N O T A T I O N

We then handed each student Table 1, shown below, and asked them to figure outthe heights of Maria and Leslie in “Story 1” namely, a case in which Tom was allegedly34 inches tall. (Actually, the value T-4 was not explicitly written into the Table but wehelped several of the students with this part so that they had at least two examples towork with; their job was then to determine the notation for Leslie in Story 4.) The tableitself was possibly a new form of representing mathematical information for the childrenand so it is not surprising to us that they would need a little guidance in understandingwhat sort of information the table presented and what sort of information needed to befilled in. Within a few minutes they were relating the information in Story 1 to theinformation given before and most students were able to determine heights consistentwith the relative information.

One might wonder why we did not provide the table at the beginning of the class sothat the children could work with specific values. In our earlier work, however, we hadnoticed that the students were often inclined to perform hasty numerical computationswithout reflecting upon the relations among the quantities at hand. By initially treatingthe heights as unknowns, we had been able to contain their urge (somewhat) tocompute before thinking about the relations among the quantities. Now that we hadachieved our basic purposes, we turned to the instantiated values as a means of varyingthe quantities while maintaining certain properties (the differences in heights) invariant.We named the table “Table of Possible Values” to emphasize the fact that we weredealing with hypothetical values. This, we hoped, would reduce their puzzlement, werethey to have tried to reason out story 2 having previously taken the values of story 1 as"the true values".

From a certain point of view, the stories contradict each other. For example, if story1 has correct information, then the information in story 2 cannot simultaneously becorrect, and so on. Getting this straight is part of the challenge of thinking hypotheticallyand thinking about the quantities as having variable quantities.

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F i g u r e 6 : F o u r " s t o r i e s " a b o u t t h e c h i l d r e n ' s h e i g h t s .

Most children had time to complete only the first two stories. But that was enough forthem to realize that the initial problem information could involve more than one set ofpossibilities, something that had not been clear when they were trying to make theirdrawings or enactments of the heights. When we later discussed the answers in thetable in front of the whole class, we tried to see whether they appreciated this by asking,“Which story is true?” By then all seemed to be comfortable with the idea that none wasnecessarily true. Melissa drew attention to the hypothetical nature of the task bypointing out that the table “says what if….”

Kevin worked through each of the examples with no difficulty. The value, T-4, wasworked out with some scaffolding from one of the researchers. The result, T+2, was hisown production. His explanation of "T+2" follows in transcript and clip 4, below.

What if…

Story 1

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F i g u r e 7 : K e v i n ' s w o r k i n g o u t o f v a r i o u s s c e n a r i o s ( s t o r i e s ) , e a c h c o n s i s t e n t w i t h t h e o r i g i n a l i n f o r m a t i o n .

T r a n s c r i p t 4 : A n e x p l a n a t i o n o f "T + 2 "

T A L L + 2 I N C H E S

Kevin: Oh, OK, Tall.

Tom's taller than Maria by 4 inches,

and Tall plus two equals…

Leslie's taller than Tom by two inches (writing T+2 for Leslie).

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C l i p 4

In a private discussion, Nathia and Jennifer were also able to conclude that Leslie’sheight must be T+2. When they attempted to explain their reasoning to the rest of theclass they suggested that the letter T under Tom could stand for “tall” or “ten (inches)”.We mentioned that we were actually using it to stand for “whatever Tom’s height was.”Given the fact that the class was ending it would have been pointless to require that allstudents understood how letters represented variable quantities in Story 4. We weresatisfied for the moment that we had been able to initiate discussion of what would be agradual familiarization with variables (here in the sense of multiple possible values) andhow they are expressed.

A L G E B R A I C N O T A T I O N B E C O M E S L E S S M Y S T E R I O U S

Approximately one month later we interviewed each of the children regarding relativedifferences problems. By and large, we were surprised by how they had becomefamiliar and confident with such problems. The following discussion with Jennifer andMelissa followed an example quite similar to the one discussed in class (about relativeheights). Melissa's characterization of x as standing for a "surprise number" suggests

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that they are thinking about letters as standing for a single, determined value. This isconsistent with Jennifer's point that "everyone in the world has a height". However, atthe end of the session they seemed to accept the idea that x could stand for each andevery value (of money) that we had worked with over the session. Could it be thatthinking in terms of unknowns is a natural stepping stone to thinking in terms of variables(as we suspect) or are these fundamentally different ways of thinking?

T r a n s c r i p t 5 : M e l i s s a a n d J e n n i f e r d i s c u s s i n g t h e m e a n i n g o f x ( o n e m o n t h a f t e r t h e c l a s s r e f e r r e d t o ) .

S U R P R I S E N U M B E R S

David: ...x's there? What do you, what do you think that was about? I mean,why were we saying, why were we using x's?

Jennifer: You might think like the x is Alan's height and it could be any height.

Melissa: Uh huh.

David: So it could be any height?

Melissa: Yeah.

David: And... and what about Martha's height?

Jennifer: It has to be 3 more than Alan's.

David: OK. So like if x is 40...

Melissa: Yeah.

David: What would Martha's height be?

Both: 43.

David: OK. If x was a hundred...

Jennifer: It would be a hundred and three.

David: OK, now, if I didn't know Alan's height, and I just had to say, "Well, Idon't know it so I'll just call it 'x'…"

Melissa: You could guess it.

Jennifer: You could say like, well it would just tell you to say any number.

David: Why don't I use an x and say whatever it is I'll just call it x? (umm) Doyou like that idea, or does that feel strange?

Jennifer: (Melissa seems to accept it but Jennifer does not) No, that’s strange.

David: It feels strange?

Jennifer: Cos it has to, it has to have a number. Cos... Everybody in the worldhas a height.

David: Oh... OK. Well, what if I tell you, I'll do it a little differently. I have alittle bit of money in my pocket, OK, do you have any coins, like anickel or something like that?

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Jennifer: All mine's in the bag...

David: OK, I'll tell you what: I'll take out, I'll take out a nickel here, ok. And I'llgive it to you for now. I've got some money in here (in a wallet) canwe call that x? (hmm) Because, whatever it is, it's (inaudible)...

Jennifer: You can't call it x because it has... if it has some money in there, youcan't just call it x because you have to count how many money is inthere.

David: But what if you don't know?

Jennifer: You open it up.

David: Because if you tell me its like 50 cents…yeah, but we haven't opened it yet.We will later.

Melissa: So its like a surprise.

David: Yeah. It's like a surprise.So let's call the x in this case the amount of money that I have inhere, OK?

Melissa: So its like an x, and then…

David: Now how much money do I…

Jennifer: X is probably like a nickel.

David: How much do I have here?

Both: x 5.

Jennifer: 5 cents.

David: x 5 or...?

Melissa: x five...

David: x five.

Jennifer: x plus 5

David: She's [Jennifer's] saying it a little differently.If I put this back in here, now I have x…?

Melissa: x.

David: No, I had x before. And now I'm putting it in here and I've got...

Both: x plus 5.

David: x plus 5, sure. And that…

Jennifer: The amount of money in there is… any money in there. And after..., ifyou like add five, if it was like... imagine if it was 50 cents, add fivemore and it would be 55 cents.

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C l i p 5 : M e l i s s a a n d J e n n i f e r d i s c u s s i n g t h e m e a n i n g o f x .

L I N K S T O E A R L Y A L G E B R A

We took as our goal to illustrate how problems normally viewed as sitting squarely inthe domain of arithmetic can take on an algebraic character from the very start. Thisargument has been made, in slightly different terms, by other members of the presentsymposium. We find ourselves in agreement with Kaput and Blanton's (1999) appealfor an algebrafying of the arithmetic curriculum. The move for Early Algebra would seemto represent a new way of looking at how arithmetic is taught and learned. As such, it isless about displacing curricula by new topics than in looking at time-honored topics in anew light and with a new set of attitudes. We expect that this will require a lot ofinvestment in teacher development and, as we tried to suggest in the present paper, animportant role for classroom research that has both an applied and basic character,where researchers and practitioners work closely together. This latter idea has beenechoed by all the contributors to this symposium at one time or another as well as ourcolleagues in mathematics education.

The CGI (Carpenter and Levi, 1991 ) work presented here highlights how much canbe achieved by focusing on written notation itself and the playful ways in which childrendelight in finding written expressions that capture generalizations they have noticedabout numbers. For those of us who have often looked upon symbol manipulation asinherently meaningless activity, it leaves a lot to ponder about. Written symbols are at

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times meaningless, but at other times, and apparently from a very early age, they maybe treated as conceptual objects, the meaning of which derives in part from children'sknowledge of number and in part from their sense of syntactical and logical coherence.Any attempt to evoke the algebraic character of arithmetic will have to consider thefundamental role of natural and written language, both of which are symbol systems.

As we prepared this report we came across a notable, earlier work by the discussant(Thompson, 1991) that shares several of the premises of our own work and revealssome similar findings. In short, Thompson found that 5th grade students were stillstruggling with some types of additive differences. The problems he used, with theexception of some given on day 1, were admittedly more complex than those used in ourstudy; many entailed second-order additive differences (differences betweendifferences). Nonetheless, the mistaking of, for example, a second-order difference(how much more brother A towers over his sister than brother B towers over his sister)with a first order difference (how much taller brother A is than sister A) echos the kindsof confusions our students exhibited when first presented with word problems.

We suspect that such difficulties are only partly about "reading carefully". As wenoted in Clip 1, students continued to insist that Maria was 6 inches tall even thoughthey very carefully read that "Maria is 6 inches shorter than Leslie". It seems thatparsing the writing notation is part of a larger story. This larger story seems to involvestudents' conceptualizations about how quantities can be formed from and broken apartfrom other quantities. This sort of analysis suggests that we we also take into accountnon-linguistic, spatial representations as part of the symbolic repertoire that comes intoplay in the emergence of early algebra. This is where other representations of aparticularly visual character, tables, vector diagrams, and free-form drawings take onimportance. If algebraic knowledge ultimately rests upon intuitions and knowledge aboutphysical quantities--if for example, variable quantities are a gateway to algebraicvariables--then we will need studies documenting the issues that children are dealingwith. There is already a good deal of evidence that a mismatch between operations onquantities and written notation lies at the heart of many of students' difficulties withmultiplicative structures and rational number concepts (e.g., Carraher, 1996). Our workwith third graders suggests that there are similar issues to be worked out regarding themapping between written notation, including notation for equations and for numbersthemselves, and their sense of what is implicit regarding the relations among quantities.

The key to viewing arithmetic through the lens of algebra consists in recognizing thesignificance of physical quantities (and actions and events with quantities) in children’sthinking as forerunners for the concepts of mathematical variable and functions.

Children commonly work with unknown quantities in arithmetic but largely as thedesired end point of a sequence of computations. Traditional wisdom holds that aproblem should contain the operations, which, if performed, will lead to an answer for thedesired unknown value. However, this only works for the most contrived andstereotyped of situations (which typical grade school textbooks abound in). A simplestatement such as “Tom is 4 inches taller than Maria” suggests to many children theoperation of adding 4, yet Maria’s height ought to be less than Tom’s height. Theadditional fact that Tom’s height is not given can perplex students expecting toencounter information in the problem given in the order that it need take in thecomputational solution. A problem such as this requires far more than simply inverting

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operations, e.g., using subtraction rather than addition. It requires taking a strikinglydifferent view of mathematical expressions. To the child heavily schooled in arithmeticas computation, a plus sign evokes the physical action of adding, giving, joining; a minussign evokes the idea of losing, spending, diminishing (Carpenter & Moser, 1982). Thereis nothing inherently wrong with this belief, and surely children are going to begin fromsuch beliefs, but they will not get very far in the vast sea of problems for which no suchphysical actions can be easily identified (see, for example, Meira, 199x). The simpleidea of a comparison (“Sally is 5 inches taller than Jamie”) provides a case in point. Aswe noted, young children can, with encouragement, use continuous visual diagrams torepresent the relative heights of children. But a problem of this sort requires isolating apart or interval of these drawn quantities as representing the additive differencesbetween two children’s heights. This requires treating the height difference as distinctquantity in its own right. The job of identifying this new conceptual entity can beapproached through attempting to draw it. But children must also be able to talk aboutsuch relational entities. As the brilliant work by Bodanskii (1991) and others has shown,this is one place where algebraic notation can play an important role. If x is thedifference between quantities A and B, and B is greater than A, then it is true that A = B-x. It is also true that A + x= B. And x is equal to B-A. From a traditional arithmeticalstandpoint, these are three different statements telling us to “do” different computations.From an algebraic viewpoint, they are merely different ways of describing the samerelations.

Algebra knowledge is not always grounded in thinking about quantities. As Kaput(1995) has noted, algebra is not always about generalizing and formalizing patterns andconstraints; there is a very legitimate sense in which algebra can be viewed as thesyntactically-guided manipulation of formalisms. And as Resnick (1985) noted, there isa point where one can forget about the situations that gave rise to the algebra andextend one’s knowledge within the rules of the algebraic symbolic system without havingto return to the situations that originally gave meaning to the expressions.

There is good reason to honor the goal of having students treat algebra as a self-contained system for manipulating symbols and deriving implications from axioms,givens and rules of inference. But this is certainly not an ideal point of departure. Formost students, what they know about physical quantities and their interrelations willprovide a rich starting point for algebraic understanding. We can help by seeing bringingout the algebraic character of arithmetic and helping familiarizing students with some ofthe representational tools of algebra well—notation, tables, and diagrams—well beforealgebra enters the curriculum as a formal topic of study.

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